Determining the Effective Parameters of Metamaterials By Jonathan Woodley A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto Copyright © 2012 by Jonathan Woodley Determining of the Effective Parameters of Metamaterials Jonathan Woodley Doctor of Philosophy Department of Electrical and Computer Engineering University of Toronto, 2012 Abstract In this dissertation the proper determination and allowable signs of the effective parameters of metamaterial structures will be examined. First, a method that was commonly used to determine the presence of a negative index of refraction will be discussed. It will be shown that this method, which relies on the appearance of transmission peaks in the region where the real parts of the effective permittivity and permeability are expected to be negative, does not provide sufficient evidence that a negative index exists. Two alternate methods will then be presented that can be used to properly determine the sign of the index. Then, the form of the index in media that exhibit backward wave propagation will be examined from a purely three dimensional wave propagation point of view. It will be shown that in an isotropic medium backward wave propagation requires that the index be negative and in an anisotropic medium it requires that the index be negative along at least one of the three principal axes. In short, the necessary and sufficient condition for the negative index of refraction is the existence of the backward wave. Next, a technique commonly used to retrieve the effective parameters in metamaterials from transmission and reflection data will be considered. It will be shown that this retrieval technique can lead to unphysical claims that the imaginary parts of the effective permittivity or permeability can be negative even though the medium remains passive. By comparing the effective parameters obtained analytically and from the retrieval technique it will be shown that these unphysical claims are the result of error in the numerical simulations. The concepts of causality and analyticity will also be discussed by considering the Lorentzian model and it will be shown that this model does not allow the imaginary parts of the permittivity or permeability to be negative in the metamaterials consisting of split ring resonators and split wires. ii Acknowledgements I would like to thank my supervisor Professor Mojahedi for his advice and guidance. I would also like to thank my fellow graduate students for all the many helpful conversations and walks. Finally, no acknowledgments section would be complete without a mention of the author’s family. I would like to thank them above all else. iii Table of Contents Chapter 1. Introduction and Background......................................................................1 1.1 Introduction........................................................................................................1 1.1.1 Veselago and Left-Handed Media ......................................................1 1.1.2 Negative Refraction ............................................................................3 1.1.3 Propagation in a Negative Index Medium ..........................................6 1.2 The First Negative Index Metamaterials............................................................7 1.2.1 The Array of Wires .............................................................................7 1.2.2 The Array of Split Ring Resonators....................................................9 1.2.3 The First Negative Index Medium....................................................11 1.2.4 Experimental Verification of Negative Refraction...........................13 1.3 Conclusion and Thesis Goals...........................................................................15 1.4 Chapter Summary ............................................................................................16 Chapter 2. Left-Handed and Right-Handed Metamaterials.......................................21 2.1 Introduction......................................................................................................21 2.2 The Transmission Magnitude ..........................................................................23 2.3 The Transmission Phase ..................................................................................26 2.4 Dispersion Diagrams........................................................................................29 2.5 The Effects of Unit Cell Size...........................................................................32 2.6 Conclusion .......................................................................................................34 Chapter 3. Backwards Waves........................................................................................37 3.1 Introduction......................................................................................................37 3.2 “Perfect” Backwards Waves............................................................................38 3.3 Backward Wave Transmission Line ................................................................43 iv 3.4 “Imperfect” Backward Waves .........................................................................44 3.5 The Requirements of Dispersion IN negative Index Metamaterials................46 3.6 Anisotropic Media ...........................................................................................47 3.7 Conclusion .......................................................................................................55 Chapter 4. The Retrieval Technique.............................................................................57 4.1 Introduction......................................................................................................57 4.2 The Retrieval Technique..................................................................................58 4.2.1 An Array of Split Wires....................................................................61 4.2.2 An Array of Split Ring Resonators...................................................65 4.2.3 Continuous Strip Wires and Split Ring Resonators..........................68 4.3 Conclusion .......................................................................................................72 Chapter 5. The Imaginary Parts of the Effective Permittivity and Permeability.....75 5.1 Introduction......................................................................................................75 5.2 The Lorentzian Model for Metamaterials........................................................77 5.3 An Array of Dielectric Spheres........................................................................81 5.3.1 Mie Theory........................................................................................82 5.3.2 Numerical Simulations......................................................................89 5.4 Discussion........................................................................................................90 5.5 Examining the Sensitivity of the Retrieval Technique ....................................96 5.6 Retrieval on a Dispersionless Dielectric Slab .................................................99 5.7 Other Homogenization Techniques ...............................................................103 5.8 Conclusions....................................................................................................106 Chapter 6. Conclusion ..................................................................................................111 List of Contributions......................................................................................................114 v Appendix A. The 6 Family Model ...............................................................................115 A.1 Introduction...................................................................................................115 A.2 Symmetry Operations ...................................................................................116 A.3 The 6 Families...............................................................................................117 A.4 The 6 Families in the 4-Quadrant Model......................................................121 A.5 Conclusion ....................................................................................................127 Appendix B. The Derivation of Heat Evolution .........................................................129 B.1 Heat evolution of Non-Monochromatic Waves ............................................129 B.2 Heat Evolution of Monochromatic Waves....................................................131 Appendix C. Mie Theory and the Array of Dielectric Spheres ................................133 C.1 Introduction ...................................................................................................133 C.2 Field patterns resulting from the sphere resonances .....................................133 C.3 Obtaining the expressions for the effective permittivity and permeability...137 vi List of Tables Chapter 3. Backwards Waves 3.1 Index seen by different polarizations in the one- and two-sheeted hyperboloids........................................................................................................52 Appendix A. The 6 Family Model A.1 Signs of u and v in each of the 6 Families......................................................121 A.2